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Second order differential operators and Dirichlet integrals with singular coefficients. I: Functional calculus of one-dimensional operators. (English) Zbl 0653.35034
Let a,c be piecewise $$C^ 1$$, $$C^ 0$$ functions on $${\mathbb{R}}$$ respectively. The authors study the parabolic Cauchy problem $\partial u/\partial t=Lu\quad if\quad t>0,\quad u|_{t=0}=0,$ where $$Lu=(1/c^ 2)\partial_ x(1/a$$ $$2)\partial_ xu)$$ (and also Cauchy problems for wave or Schrödinger equations). Explicit formulas for the heat kernel are given. The case where L is a spherically symmetric elliptic operator in $${\mathbb{R}}^ 3$$ is also treated.
Reviewer: P.Godin

##### MSC:
 35K15 Initial value problems for second-order parabolic equations 35L15 Initial value problems for second-order hyperbolic equations 35R05 PDEs with low regular coefficients and/or low regular data 35C05 Solutions to PDEs in closed form